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Theorem 1st2nd 6334
Description: Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)
Assertion
Ref Expression
1st2nd  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )

Proof of Theorem 1st2nd
StepHypRef Expression
1 df-rel 4827 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
2 ssel2 3288 . . 3  |-  ( ( B  C_  ( _V  X.  _V )  /\  A  e.  B )  ->  A  e.  ( _V  X.  _V ) )
31, 2sylanb 459 . 2  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  e.  ( _V  X.  _V ) )
4 1st2nd2 6327 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
53, 4syl 16 1  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2901    C_ wss 3265   <.cop 3762    X. cxp 4818   Rel wrel 4825   ` cfv 5396   1stc1st 6288   2ndc2nd 6289
This theorem is referenced by:  2ndrn  6336  1st2ndbr  6337  elopabi  6353  cnvf1olem  6385  ordpinq  8755  addassnq  8770  mulassnq  8771  distrnq  8773  mulidnq  8775  recmulnq  8776  ltexnq  8787  fsumcnv  12486  cofulid  14016  cofurid  14017  idffth  14059  cofull  14060  cofth  14061  ressffth  14064  isnat2  14074  nat1st2nd  14077  homadmcd  14126  catciso  14191  prf1st  14230  prf2nd  14231  1st2ndprf  14232  curfuncf  14264  uncfcurf  14265  curf2ndf  14273  yonffthlem  14308  yoniso  14311  dprd2dlem2  15527  dprd2dlem1  15528  dprd2da  15529  2ndcctbss  17441  utop2nei  18203  utop3cls  18204  caubl  19133  rngoi  21818  drngoi  21845  nvop2  21937  nvvop  21938  nvop  22016  phop  22169  cvmliftlem1  24753  heiborlem3  26215  isdrngo1  26265  iscrngo2  26301
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-iota 5360  df-fun 5398  df-fv 5404  df-1st 6290  df-2nd 6291
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